Estimating the perceived quality of an audio signal is critical for many multimedia and audio processing systems. Providers strive to offer optimal and reliable services in order to increase the user quality of experience (QoE). In this work, we present an investigation of the applicability of neural networks for non-intrusive audio quality assessment. We propose three neural network-based approaches for mean opinion score (MOS) estimation. We compare our results to three instrumental measures: the perceptual evaluation of speech quality (PESQ), the ITU-T Recommendation P.563, and the speech-to-reverberation energy ratio. Our evaluation uses a speech dataset contaminated with convolutive and additive noise, labeled using a crowd-based QoE evaluation, evaluated with Pearson correlation with MOS labels, and mean-squared-error of the estimated MOS. Our proposed approaches outperform the aforementioned instrumental measures, with a fully connected deep neural network using Mel-frequency features providing the best correlation (0.87) and the lowest mean squared error (0.15).